Verification methods and agronomic enhancements for carbon removal based on enhanced rock weathering

ABSTRACT

The present disclosure relates to methods of verifying enhanced rock weathering using immobile trace elements found within a mineral amendment. Further disclosed are mineral amendments that enable enhanced rock weathering.

CROSS REFERENCE TO RELATED APPLICATIONS

The present applications claims the priority benefit of U.S. ProvisionalPatent App. Ser. No. 63/322,672, filed Mar. 23, 2022; U.S. ProvisionalPatent App. Ser. No. 63/289,395, filed Dec. 14, 2021; and U.S.Provisional Patent App. Ser. No. 63/213,398, filed, Jun. 22, 2021; eachof the foregoing of which are incorporated herein by reference in theirrespective entireties.

BACKGROUND

The last several years have witnessed a maturation of carbon marketsfrom lightly scrutinized voluntary markets largely serving to providepositive marketing collateral to familiar consumer brands to rigorouscompliance markets with proof-of-performance requirements ofteninvolving government and quasi-governmental regulators. There have alsobeen a number of comprehensive proposals for decarbonization of theentire US economy, combining energy production, transportation, cementproduction, and agriculture. These proposals have included carbondioxide removal (CDR) techniques as an essential component of thedecarbonization plan, with a supportive ecosystem of science, policy,and project evaluation criteria. Within the diverse scope ofdecarbonization efforts, including renewable energy, biofuel,nature-based solutions, and more technical CDR techniques like directair capture (DAC), The Oxford Principles have developed a taxonomy forcategorizing diverse decarbonization strategies. This taxonomydistinguishes between avoided emissions (for instance, conversion offossil fuel power to renewable power) and negative emissions (forinstance direct air capture); distinguishes between avoided emissionsthat require storage (for instance carbon capture from a point sourceand sequestration into the ground) and those that do not require storage(the conversion to renewable energy); and the duration of the storage ifso required (for instance, short lived forest carbon sequestration fromdelayed harvests versus storage of captured carbon in geologicalformations).

A number of for-profit and non-profit organizations have emerged thatemploy the Oxford Principles to evaluate proposals, particularly aroundthe rigorous quantification across categories (Table 1). This, inessence, is the scorecard that projects will have to compete on to findacceptance in the industry. All other things equal, cost ends up beingthe primary driver, but quality is also a key consideration. Thus,historically buyers have been drawn to nature-based solutions that costbetween $5-20/tCO₂ e (e.g., Corteva Carbon, Indigo Carbon), versusdirect air capture projects that cost >$500/tCO₂ e (e.g., ClimeWorks).However, there are challenges to these nature-based solutions: recentlythese nature-based solutions have been subject to intense scrutiny for“gaming” the rules or otherwise failing to deliver prospective rewards.Among the technological solutions, there are a different set ofchallenges: rigor has been strong, but the reduction in priceproportional to increase in deployment has been slow, and the ultimateprice of DAC is not expected to drop below $150/tCO₂e (Keith et al 2018,https://doi.org/10.1016/j.joule.2018.05.006). In light of thesechallenges, we have developed a technology around “enhanced rockweathering” (ERW), in which silicate minerals are weathered in acidicsoil solution, thus driving the uptake of additional CO₂ into dissolvedinorganic carbon (DIC) in the soil solution. Our experimental work, lifecycle analysis, and techno-economic modeling have indicated that ERWprovides the permanence, additionality, and rigorous quantification of aDAC project, the large volume of a carbon capture and storage (CCS)project, and the unit economics of a nature based solution, thuscreating a new tool to meet net-zero goals.

Details of Enhanced Rock Weathering

In “enhanced rock weathering” (ERW), silicate minerals are weathered byCO₂ from the ambient air which has been dissolved into water, reactingto produce dissolved inorganic carbon that is ultimately stored in theocean as depicted in FIGS. 1 and 2 .

Background Geochemistry

The role of silicate rock weathering in maintaining the CO₂ balance ofthe atmosphere has been recognized for decades, first outlined byrenowned chemist Harold Urey in the 1950s. The basic premise of the Ureyreaction is that continental collisions release CO₂ to the atmospherefrom volcanoes, and bring Mg- and Ca-bearing silicate rocks to thesurface. The rainwater (H₂O) that falls upon these rocks is mildlyacidic, as atmospheric CO₂ has dissolved in it and formed carbonic acid(H₂CO₃). For forsterite, the weathering reaction takes place followingthe form:

Mg₂SiO₄+4H₂O+4CO₂→2Mg²⁺+4HCO₃ ⁻+4H₄SiO₄

In this reaction, one mole of forsterite consumes four moles of CO₂, sotwo negatively charged bicarbonate HCO₃ ⁻ are created for every onedivalent Mg′ weathered. Given the molecular weight of forsterite (140g/mol) and the molecular weight of CO₂ (44 g/mol), weathering one metricton of forsterite removes 1.25 metric tons of CO₂ from the atmosphere.In general, such a formula can be used to define the “mineral potential”of a silicate based on its elemental composition:

$\begin{matrix}{{{MP} \equiv \frac{t{CO}_{2}e}{tO{re}}} = {{\frac{MW_{{CO}_{2}}}{100\%}.\left( {\frac{{Mg}\%}{MW_{Mg}} + \frac{{Ca}\%}{MW_{Ca}}} \right)}*V}} & \left( {{Equation}1} \right)\end{matrix}$

where V is the valence of the cation (2 for Mg and Ca) and MW is themolecular weight.

The rate of weathering is determined by the surface area of the mineral,the acidity (pH) of the soil solution, the temperature of the solution,the availability of CO₂ reagent in solution, and the rate of removal ofthe reaction products by water. One of the key insights into thepotential of “enhanced” rock weathering is that the rate of reaction,and thus CO₂ removal, can be greatly accelerated by increasing thesurface area by pulverization into a fine powder (e.g., less than 100um), incorporating into an acidic environment (e.g., pH less than 6)with abundant CO2 present, and with steady water flux to remove reactionproducts to maintain acidity. While this formula defines the potentialamount of CO₂ that may be removed by weathering, it does not speak tothe rate; ancient rock formations testify that the rate of weatheringcan be extremely slow.

The relationship between silicates (like Mg₂SiO₄) and dissolvedcarbonates (like HCO₃)) is not necessarily intuitive, as the carbonatesystem of water involves a number of coupled reactions. Atmospheric CO₂dissolves into water according to an exchange coefficient:

CO_(2(aq))═H₂CO₃═K_(CO) ₂ *P_(CO) ₂

Technically pCO₂ is “fugacity” that represents its activity, but, asused herein, it is more or less equivalent to its partial pressure, andK_(CO) ₂ is the Henry's law coefficient that determines the aqueous CO₂in equilibrium with the atmosphere. This aqueous CO₂ in turn hydrateswith H₂O to become carbonic acid (H₂CO₃), which dissociates to becomebicarbonate (HCO₃ ⁻) and carbonate (CO₃ ²⁻):

H₂CO₃↔HCO₃ ⁻H⁺

HCO₃ ⁻↔CO₃ ²⁻H⁺

These equilibrium reactions are defined by K1 and K2, the first andsecond carbonate system dissociation constants. pK1 and pK2 are about5.9 and about 9.4 at STP, so greater than 99% of the charge in thedissolved carbonates is HCO₃ ⁻.

The carbonates are the largest constituents of total alkalinity, whichis defined as the charge imbalance between weak acids (proton acceptors)minus proton donors:

TA=[HCO₃ ⁻]+2·[CO₃ ²⁻]+[OH⁻]−[H⁺]

There is an alternative definition of total alkalinity as the chargeimbalance between conserved cations and conserved anions:

TA=[Na⁺]+2[Mg²⁺]2[Ca^(2+]+)[K⁺]+ . . . −[Cl⁻]−2[SO₄ ²⁻]−[NO₃ ⁻]

These two expressions are always equal (i.e. the charges balance). Thismeans that an added Mg or Ca into soil solution will increase in HCO₃ tobalance the charge. We describe an analytical solution to compute howmuch carbon is taken up per unit of additional Mg or Ca. If we firstdefine DIC as the sum of H₂CO₃, HCO₃ ⁻, and CO₃ ²⁻, and make use of theequilibrium equations above (defining H⁺ as “h” and H₂CO₃ as “s”):

${DIC} = {s \cdot \left\lbrack {1 + \frac{K_{1}}{h} + \frac{K_{1}K_{2}}{h^{2}}} \right\rbrack}$

And formulate TA using the same convention:

${TA} = {{s \cdot \frac{K_{1}}{h}} + {s \cdot 2 \cdot \frac{K_{1}K_{2}}{h^{2}}} + \frac{K_{w}}{h} - h}$

With these definitions in place, we can develop an estimate of dDIC/dTA.First, we compute the derivative dTA/dh:

$\frac{dTA}{dh} = {{{- s} \cdot \left( {\frac{K_{1}}{h^{2}} + {4 \cdot \frac{K_{1}K_{2}}{h^{3}}}} \right)} - \frac{K_{w}}{h^{2}} - 1}$

Next we compute the derivative dDIC/dh:

$\frac{dDIC}{dh} = {{- s} \cdot \left( {\frac{K_{1}}{h^{2}} + {2 \cdot \frac{K_{1}K_{2}}{h^{3}}}} \right)}$

Finally we multiply dDIC/dh by the inverse of dTA/dh to calculatedDIC/dTA:

$\frac{dDIC}{dTA} = {\frac{dDIC}{dh} \cdot \frac{dh}{dTA}}$

At the pH found in soils, each Mg or Ca is matched by two carbon atoms.In all circumstances, raising alkalinity results in increased pH. Thisis particularly important in marine settings, where ocean acidificationfrom increased atmospheric CO₂ can be ameliorated by this export ofalkalinity from land.

The potential for ERW as a commercial enterprise is limited by somefundamental issues:

-   -   A. The total amounts of mineral transformation and carbon        removal should be verifiable empirically. The verification        methods described below may be low cost at scale, can be        performed at arm's length by a 3rd party, and may have        safeguards to eliminate fraud.        Additionally, the commercial potential for ERW could be enhanced        by a number of different features. Embodiments may include:    -   B. Processes and modifications of the engineered mineral product        that enhance the agronomic performance and ecosystem co-benefits        of the engineered mineral.    -   C. Processes and modifications of the engineered mineral product        that control or enhance the mineral dissolution rate of the        engineered mineral in soil environments and thus the rate of        removal of CO₂.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a drawing illustrating the enhanced rock weathering cycle.

FIG. 2 is a graph depicting the effect of comminution on dissolutionkinetics for a silicate.

FIGS. 3A to 3D are graphs depicting the kinetics for a particle sizedistribution of a pulverized silicate with median particle size of80-100 um which closely approximates the alkalinity release dynamics foraglime.

FIG. 4 is a drawing illustrating the general scheme for a verifiedenhanced rock weathering cycle.

DETAILED DESCRIPTION

Summary: The following described methodologies can establish one versionof a verification scheme that will demonstrate the transformation of theapplied rock material and the subsequent carbon removal into securegeologic reservoirs as evidenced by observations collected from thesoil. One set of methods (A) can measure the production of free ionsfrom the applied material and the transport of those ions outside of acontrol volume as a direct measure of the amount of weathering and thustotal carbon dioxide equivalent isolated from the atmosphere.Furthermore, as the elemental inputs into the soil induced by enhancedrock weathering can have significant effects on the soil geochemistry,methods in (B) describe systems and versions that can enhance theagronomic performance of the soil amendment and increase ecosystemco-benefits. Lastly, methods in (C) elaborate on versions that controland enhance the mineral dissolution rate, which increases the financialperformance of enhanced rock weathering technology in the marketplace.

A. Systems and Methods for Monitoring and Verification:

-   -   Described below are several example methodologies for        quantifying the rate and extent of mineral transformation and        carbon removal, which are referred to as “verification        methodologies”.    -   1. Example Verification methodology 1:        -   a. A cation exchange resin and/or an anion exchange resin            are packaged into a physical embodiment (such as a 5 cm            diameter tube, 5 cm in length, placed 30 cm below the soil            surface) that allows vertical transport of fluids but not            horizontal transport.        -   b. Commercial ion exchange resins are pre-equilibrated in            such a way that their selectivity is high for the relevant            ions, but their capacity is also sufficiently high such that            saturation is minimized. This technology may allow for            effective deployment for long periods of time, allowing            passage of up to 4000 mm of moisture through the tube,            without becoming saturated with respect to the ions of            interest.        -   c. The ions of interest are restricted to divalent cations            (Mg²⁺, Ca²⁺), carbonate species (HCO₃ ⁻ and CO₃ ²), and            silicic acid (H₄SiO₄), important weathering products of            preferred mineral soil amendments, such as ultramafic rocks,            blast furnace slag, and other naturally occurring or            industrial silicate minerals with high Mg and Ca content.        -   d. Three pre-equilibrated tubes are emplaced fully under the            soil below a control depth (e.g., 30 cm, a typical depth of            cultivation). This may be done by removing the top 30 cm            soil with a shovel or auger, then gently placing the tube in            the created void space. Following a certain period of time            post-surface application of mineral soil amendment (e.g., 9            to 12 months after application), the tubes are retrieved,            and the ions captured within the resin exchanges are            measured using standard methodologies, such as ICP-OES and            ICP-MS based concentration measurements. Three tubes are            used to determine standard deviation error in triplicate.        -   e. By the principle of charge balance and known            thermodynamic reactions taking place in the top 30 cm of            soils, the amount of carbonate can be inferred purely from            the measurement of cations, under certain assumptions and            auxiliary geochemical data. By the addition of more specific            anion exchange resins, such as styrene-divinylbenzene, these            assumptions can be avoided to get a more precise answer. The            increase in cation concentration in the subsurface soil pore            water allows for the stoichiometric determination of carbon            removal from the gas phase (ambient air).        -   f. The total carbon dioxide removed during the deployment            period may be estimated as equal to the molecular mass of            CO₂ (44 g/mol) multiplied by the sum of the number of moles            of carbonate and two times the number of moles of            bicarbonate, multiplied by the area on which mineral was            applied to the field, divided by the cross-sectional area of            the tube.        -   g. Direct measurement of aqueous bicarbonate and carbonate            species, which are the most readily available forms of            carbon induced by a gas exchange of carbon dioxide with the            soil pore water, may be conducted using the specified anion            exchange resin above. The expression as described in            clause (f) explains the calculation used to convert this            direct measurement of carbon to an absolute value of carbon            dioxide removed during a specified application period.        -   h. Verification methodology 1 has the advantage that it can            provide a direct measure of carbon flux and employs a            measurement technique that is mature.        -   i. However, potential disadvantages may include the need for            manufacturing (and its attendant demand for working            capital); devices may occasionally be defective; emplacement            of a device may alter the flow paths of water, which in turn            alters the inferred ion fluxes; emplacement of a device may            depend on the user; the knowledge of the location of            emplacement may invite manipulation by a stakeholder            involved in a carbon transaction; and others.    -   2. Example Verification methodology 2:        -   a. The principle behind Verification methodology 2 is that            minerals applied for purposes of enhanced rock weathering            contain, in addition to the elements outlined above that            participate in the weathering reaction (i.e., magnesium,            calcium, iron, silicon, oxygen, hydrogen), additional            elements in trace amounts. These additional trace elements            (TEs) might include rare earth elements (REEs), rare metals            (RM), other transition metals (TMs), or a combination            thereof. Unlike the primary weathering products that are            readily dissolved in solution and lost by leaching, some TEs            are strongly bound to mineral and biological surfaces and do            not readily leach from the soil control volume (e.g., the            top 10 to 30 cm of soil), and are not removed by plants at            the concentrations present in our applied rock. These            strongly bound trace elements are referred to herein as            immobile trace elements (ITEs). Thus, a measure of            cumulative cation flux and carbon removal can be computed by            comparing the ratio of the lost weathering products, such as            magnesium, to ITEs, after accounting for background            concentrations of ITEs in the initial soil.        -   b. As used herein, rare earth elements include scandium            (Sc), yttrium (Y), lanthanum (La), cerium (Ce), praseodymium            (Pr), neodymium (Nd), samarium (Sm), europium (Eu),            gadolinium (Gd), terbium (Tb), dysprosium (Dy), holmium            (Ho), erbium (Er), thulium (Tm), ytterbium (Yb), lutetium            (Lu) or a combination thereof. As used herein, rare metals            include beryllium (Be), cesium (Cs), gallium (Ga), germanium            (Ge), hafnium (Hf), niobium (Nb), rubidium (Rb), tantalum            (Ta), zirconium (Zr), or a combination thereof. As used            herein, transition metals include nickel (Ni), chromium            (Cr), and zinc (Zn) among others and may include a            combination of transition metals.        -   c. An approach based on ITEs depends on the degree of            immobility of the element in soil environments, the time            horizon after which the ITEs will be measured to estimate            carbon removal, the abundance of ITEs in both the soil and            the mineral added, and the analytical chemistry used to            measure this abundance. An analytical technique that has low            detectability thresholds for slightly mobile elements could            be sufficient for relatively abundant (e.g., parts per            thousand to parts per million) elements over short time            horizons (e.g., weeks to months). On the other hand, a            different analytical technique could be used over longer            time horizons, which in turn could involve quantification of            very immobile elements that are present in much lower            abundance (e.g., parts per billion to parts per trillion).            The approach may be chosen based at least in part on cost,            which ITEs are actually conserved, over which time horizons,            as well as the performance of the analytical performance            used.        -   d. The relative proportions of ITEs in a sample (rock or            soil) constitute a type of unique fingerprint of the            material. Because the cost of pure ITEs is large, and the            analytical chemistry is not widely available, there is a            significant barrier to engineering a fraudulent sample to            reproduce the native ITE fingerprint of a mineral soil            amendment. Thus, it would be challenging for a stakeholder            in a carbon exchange transaction to generate a result that            produces the anticipated result (a high or low amount of            carbon removal) while also matching the ITE fingerprint            generated by a bona fide sample. This is a contrast with            measurement schemes for soil organic carbon, in which the            landowner or other interested party has an information            advantage in terms of where or when to sample, which could            be used to achieve a particular carbon measurement objective            (high or low).        -   e. Verification Methodology 2 (VM2) has advantages over            Verification Methodology 1 (VM1) in that it avoids the need            for the use of a device, which avoids altering the soil            hydrology or physical environment in a way that would impact            the observed elemental analysis.        -   f. VM2 may have an advantage over VM1 in not requiring            tracking of a specific location for recovering the sampling            column.        -   g. VM2 does, however, include measurement techniques that            are less widely available; requires specialized instruments            and personnel; requires additional boundary conditions to            compute carbon flux (e.g., pre-application soil and rock            elemental analysis); and employs assumptions as to the            relationship between cation flux and carbon flux.        -   h. Like VM1, VM2 uses the total amount of mineral applied            and the field area, which could be measured, for example,            using a digital as-applied map commonly accompanying crops            input machinery, or truck weights before and after mineral            delivery. This will be referred to as the nominal            application rate (AR_(nominal)).        -   i. There are four main calculations in VM2. To the extent            any one of these factors is known unambiguously from other            sources, each step in VM2 could be used in isolation from            the others.            -   i. Classify, using ITE fingerprinting, whether a soil                has had a specific mineral applied, which may be                asserted by an entity wishing to make a claim;            -   ii. Calculate, using ITE fingerprinting, what the actual                mineral application rate (AR_(actual)) for a specific                soil sample was, which will necessarily differ in a                systematic or random way from the average rate for the                entire field;            -   iii. Calculate the amount of divalent cations remaining                in a control volume relative to the amount predicted by                the application rate in (ii) above;            -   iv. Calculate, using the fraction computed in (iii) and                the mineral potential (Equation 1) computed for the                feedstock identified in (i) above, the amount of carbon                dioxide removed.        -   j. In one non-limiting embodiment, the verification            methodology to classify whether the claimed mineral applied            is the actual mineral applied is as follows:            -   i. Collect a sample from mineral to be applied, place in                a secure vessel and seal with another secondary bag in                order to reach air-tight containment. Label as Mineral                Amendment.            -   ii. Prior to mineral application, collect a 20 g sample                from cultivated zone of soil (typically at around 30 cm,                such as in a range of 0 cm to 30 cm), place in a secure                vessel and seal with another secondary bag in order to                reach air-tight containment. Label as Soil A.            -   iii. Subsequent to mineral application, collect a 20 g                sample the same control zone as soil A, and store under                similar conditions. Label as Soil B. Soil B could be                sampled immediately after mineral application or at a                later time (e.g., months to years afterwards).            -   iv. Characterize the elemental composition of Mineral                Amendment, Soil A, and Soil B. Example approaches to                characterizing the elemental composition are described                in (v) and (vi).            -   v. One approach to measuring the elemental composition                is as follows:                -   1. In a laboratory setting, unseal the soil samples                    and place 10 grams of Soil A and Soil B in two                    separate beakers.                -   2. Dissolve the solids in strong acid. For example,                    add 21 mL of strong 1.0 M hydrochloric acid and 7 mL                    of strong 1.0 M nitric acid (total 28 mL acid) to                    each beaker of sterilized soil and stir rigorously                    until solids are dissolved.                -   3. Filter the sample-containing beakers first                    through a 1- to 5-micron water filter cartridge                    followed by an attached 10 to 20 cm column cation                    exchange resin at a sufficiently slow flow rate of,                    for example, 1 to 5 mL/min. This combination of a                    water filter with a cation exchange column has been                    engineered for optimal performance with soils under                    consideration. Specifically, the 5-micron water                    filter cartridge separates out any larger particles,                    allowing for the smaller particles to exchange its                    bound metals with the subsequent cation exchange                    resin.                -   4. Flush copious amounts (e.g., 100-200 mL) of 0.5                    to 1.0 M chelator such as, for example,                    ethylenediaminetetraacetic acid (EDTA) at 1-5 mL/min                    through the used filters into their respective                    filtrates. This may collect any additional metals                    that were still adhered to the filters.                -   5. Send filtrate A (resultant sample from Soil A)                    and filtrate B (resultant sample from Soil B) to an                    analytical laboratory for analysis on an inductively                    coupled plasma mass spectrometer (ICP-MS).                    Measurements may include concentrations of high ppb                    detection of REEs, high ppm detection of Mg, and                    percent mineralogical fractions of SiO2, Al2O3, and                    Fe2O3.                -   6. The ICP-MS may be set to detect magnesium                    concentrations and the following 17 elements on a                    high ppb or low ppm detection level: Yttrium,                    Scandium, Lanthanum, Cerium, Praseodymium,                    Neodymium, Promethium, Samarium, Europium,                    Gadolinium, Terbium, Dysprosium, Holmium, Erbium,                    Thulium, Ytterbium, Lutetium. Classify light rare                    earth elements (LREEs) as yttrium, scandium,                    lanthanum, cerium, praseodymium, neodymium,                    promethium, and samarium. Classify heavy rare earth                    elements (HREEs) as europium, gadolinium, terbium,                    dysprosium, holmium, erbium, thulium, ytterbium, and                    lutetium. Send an isolated rock material sample for                    ICP-MS analysis. Similar to the filtrate analyses,                    measurements may include concentrations of high ppb                    detection of REEs, high ppm detection of Mg, and                    percent mineralogical fractions of SiO2, Al2O3, and                    Fe2O3.            -   vi. Another approach to measuring the elemental                composition is as follows:                -   1. Calibrate a portable XRF instrument for                    particularly low detection limit of ITEs using a                    calibration standard (such as Bruker proprietary Geo                    calibration+custom ITE standards).                -   2. Use the portable XRF instrument to analyze the                    Rock Material and Soil A for ITEs (as identified in                    Verification methodology 2), reporting the                    instrumental error (2 standard deviations) as well.                -   3. Use the portable XRF instrument to analyze Soil B                    for ITEs, reporting the instrumental error (2                    standard deviations) as well.            -   vii. Once the above elemental composition has been                determined, calculate the difference for every element                between Mineral Amendment and Soil A, resulting in a                vector of differences v_(rock). Likewise, compute the                difference for every element between Soil B and Soil A,                resulting in a vector of differences v_(soil). To                improve performance, each element may be divided by an                individual factor, such as the detection threshold, or                the instrumental uncertainty, or the elemental                composition of a reference material. Another means to                improve performance would be to sum a subset of the                elements or several distinct subsets of elements before                computing v_(rock) and v_(soil). Another means to                improve performance would be to compute indices of these                summed subsets of elements, for example the ratio of the                light REEs to the heavy REEs, before computing v_(rock)                and v_(soil).            -   viii. Compute the dot product of v_(rock) and v_(soil).                If the value of this dot product is close to 1 (e.g.,                within a threshold value), then the Soil B is positively                classified as having the Mineral Amendment. If the value                of the dot product is less than 1 minus a threshold                value, then Soil B is negatively classified as not                having the Mineral Amendment.            -   ix. The threshold value may be determined using a                variety of means, for example the half-angle between the                dot product of v_(rock) and v_(soil) and any other known                v_(rock) and v_(soil); or using a Monte Carlo simulation                of v_(rock) and v_(soil) that accounts for known sources                of uncertainty including instrumental error, variation                in rock elemental analysis, or variation of soil                elemental analysis.        -   k. In one non-limiting embodiment, the verification            methodology to calculate the actual mineral application rate            (AR_(actual)) for a specific soil sample is as follows:            -   i. Characterize the elemental composition the elemental                composition of Mineral Amendment, Soil A, and Soil B as                above. Note that there could be different analytical                chemistry employed if, for example, rapid and                inexpensive XRF analysis was used for classification                soon after application, and slower and more costly ICP                analysis may be used to estimate application rates.            -   ii. Compute the differences in elemental composition as                above, potentially including similar performance                enhancements such as normalizing by factors specific to                each element, summing across subsets of elements, and                computing indices such as the ratio of light REEs to                heavy REEs. Such transformed and summed variables for a                sample will be referred to generically as ΣITE_(sample).            -   iii. The actual application rate of the sample can be                calculated using the following expression:

$\begin{matrix}{{ARactual} = {x\left\lbrack \frac{\left( {{\sum{ITEs}_{soilB}} - {\sum{ITEs}_{soilA}}} \right)}{\left( {\sum{ITEs}_{mineral}} \right)} \right\rbrack}} & \left( {{Equation}2} \right)\end{matrix}$

-   -   -   l. In one non-limiting embodiment, the verification            methodology to calculate the amount of divalent cations            remaining in a control volume relative to the amount            predicted by the actual mineral application rate            (AR_(actual)) as follows:            -   i. The amount of divalent cations applied (DCapplied)                can be calculated by the proven identity of the mineral                additive and the actual application rate:

DCapplied=AR_(actual)*[Mg_(mineral) Ca_(mineral)]  (Equation 3)

-   -   -   -   where divalent cations are generally restricted in this                context to Mg and Ca, and Mg_(mineral) and Ca_(mineral)                is the fractional composition of the mineral additive.            -   ii. The amount of divalent cations remaining (DCremain)                in the soil could be estimated using the same elemental                analysis as previously (if, for example, ICP-MS was                used), or determined from the same sample using a                different analysis (if, for example XRF was used and Mg                was not measured).            -   iii. The fractional progress of carbon dioxide removal                (CDR) can be estimated by the ratio of remaining                divalent cations to applied divalent cations:

$\begin{matrix}{f_{CDR} = \frac{DCremain}{DCapplied}} & \left( {{Equation}4} \right)\end{matrix}$

-   -   -   m. In one non-limiting embodiment, the verification            methodology to calculate the amount of carbon dioxide            removed from the control volume is as follows:            -   i. Assembling MP, f_(CDR), AR_(nominal) and the land                area applied, the total carbon dioxide removal is:

CDR=f_(CDR)*MP*AR_(nominal)*Area  (Equation 5)

-   -   -   n. While Verification Methodology 2 presented herein            provides a broad framework for the detection and application            of soil-bound ITEs to estimate long-term geologic carbon            drawdown, the efficacy of the approach can also be improved            through the focus on specific subsets of ITEs. Non-limiting            embodiments of such improvements are listed below:            -   i. The ITEs specified in equation 2, e.g.,                (ΣITEs_(soil B)), may in fact be represented not by the                full summation of 17 REEs but instead by the use of only                LREEs or the use of only HREEs, as defined in                clause xii. Such a calculation would mean the sum of                only LREEs in the calculation (yttrium, scandium,                lanthanum, cerium, praseodymium, neodymium, promethium,                and samarium) or the sum of only HREEs in the                calculation (europium, gadolinium, terbium, dysprosium,                holmium, erbium, thulium, ytterbium, and lutetium).            -   ii. Within the use of only LREEs for the calculation                (yttrium, scandium, lanthanum, cerium, praseodymium,                neodymium, promethium, and samarium), some LREEs may                serve as immobile elements more effectively than others.                In order to establish an acceptable average and standard                deviation for the C drawdown calculation, a subset of                the individual LREEs are selected. For example, the                subset may include three LREEs. The calculation                (equation 2) is conducted for each of the 3 LREEs                individually, and an average and standard deviation is                then reported for the CDR calculation. Among the 3                LREEs, the following triplicates are listed as                potential, non-limiting candidates of interest:                [yttrium, scandium, lanthanum], [cerium, lanthanum,                neodymium], [cerium, neodymium, samarium], [yttrium,                cerium, neodymium], [scandium, neodymium, samarium].            -   iii. Within the use of only HREEs for the calculation                (europium, gadolinium, terbium, dysprosium, holmium,                erbium, thulium, ytterbium, and lutetium), some HREEs                may serve as immobile elements more effectively than                others. In order to establish an acceptable average and                standard deviation for the C drawdown calculation, a                subset of the individual HREEs are selected. For                example, the subset may include three HREEs. The                calculation (equation 2a) is conducted for each of the 3                HREEs individually, and an average and standard                deviation is then reported for the CDR calculation.                Among the 3 HREEs, the following triplicates are listed                as potential, non-limiting candidates of interest:                [europium, gadolinium, terbium], [europium, terbium,                dysprosium], [dysprosium, erbium, ytterbium], [europium,                erbium, ytterbium], [europium, dysprosium, erbium].            -   iv. In order to establish an acceptable average and                standard deviation for the C drawdown calculation, a                subset of the individual REEs are selected. For example,                the subset may include three REEs Chart 1 represents a                non-limiting list of REE triplicates that may enhance                the efficacy of this C drawdown calculation.

Chart 1. A Non-Limiting List of 455 REE Triplicates that can be Used toEffectively Triangulate a C Drawdown Estimation

(‘Y’, ‘La’, ‘Ce’), (‘Y’, ‘La’, ‘Pr’), (‘Y’, ‘La’, ‘Nd’), (‘Y’, ‘La’,‘Sm’), (‘Y’, ‘La’, ‘Eu’), (‘Y’, ‘La’, ‘Gd’), (‘Y’, ‘La’, ‘Tb’), (‘Y’,‘La’, ‘Dy’), (‘Y’, ‘La’, ‘Ho’), (‘Y’, ‘La’, ‘Er’), (‘Y’, ‘La’, ‘Tm’),(‘Y’, ‘La’, ‘Yb’), (‘Y’, ‘La’, ‘Lu’), (‘Y’, ‘Ce’, ‘Pr’), (‘Y’, ‘Ce’,‘Nd’), (‘Y’, ‘Ce’, ‘Sm’), (‘Y’, ‘Ce’, ‘Eu’), (‘Y’, ‘Ce’, ‘Gd’), (‘Y’,‘Ce’, ‘Tb’), (‘Y’, ‘Ce’, ‘Dy’), (‘Y’, ‘Ce’, ‘Ho’), (‘Y’, ‘Ce’, ‘Er’),(‘Y’, ‘Ce’, ‘Tm’), (‘Y’, ‘Ce’, ‘Yb’), (‘Y’, ‘Ce’, ‘Lu’), (′ Y′, ‘Pr’,‘Nd’), (‘Y’, ‘Pr’, ‘Sm’), (‘Y’, ‘Pr’, ‘Eu’), (‘Y’, ‘Pr’, ‘Gd’), (‘Y’,‘Pr’, ‘Tb’), (‘Y’, ‘Pr’, ‘Dy’), (‘Y’, ‘Pr’, ‘Ho’), (‘Y’, ‘Pr’, Er′),(‘Y’, ‘Pr’, ‘Tm’), (‘Y’, Pr′, ‘Yb’), (‘Y’, Pr′, ‘Lu’), (‘Y’, ‘Nd’,‘Sm’), (‘Y’, ‘Nd’, ‘E u’), (‘Y’, ‘Nd’, ‘Gd’), (‘Y’, ‘Nd’, ‘Tb’), (‘Y’,‘Nd’, ‘Dy’), (‘Y’, ‘Nd’, ‘Ho’), (‘Y’, ‘Nd’, ‘Er’), (‘Y’, ‘Nd’, ‘Tm’),(‘Y’, ‘Nd’, ‘Yb’), (‘Y’, ‘Nd’, ‘Lu’), (‘Y’, ‘Sm’, ‘Eu’), (‘Y’, ‘Sm’,‘Gd’), (‘Y’, ‘Sm’, ‘Tb’), (‘Y’, ‘Sm’, ‘Dy’), (‘Y’, ‘Sm’, ‘Ho’), (‘Y’,‘Sm’, Er′), (‘Y’, ‘Sm’, ‘Tm’), (‘Y’, ‘Sm’, ‘Yb’), (‘Y’, ‘Sm’, ‘Lu’),(‘Y’, ‘Eu’, ‘Gd’), (‘Y’, ‘Eu’, ‘Tb’), (‘Y’, ‘Eu’, ‘Dy’), (‘Y’, ‘Eu’,‘Ho’), (‘Y’, ‘Eu’, ‘Er’), (‘Y’, ‘Eu’, ‘Tm’), (‘Y’, ‘Eu’, ‘Yb’), (‘Y ’,‘Eu’, ‘Lu’), (‘Y’, ‘Gd’, ‘Tb’), (‘Y’, ‘Gd’, Dy′), (‘Y’, ‘Gd’, ‘Ho’),(‘Y’, ‘Gd’, ‘Er’), (‘Y’, ‘Gd’, ‘Tm’), (‘Y’, ‘ Gd’, ‘Yb’), (‘Y’, ‘Gd’,‘Lu’), (‘Y’, ‘Tb’, ‘Dy’), (‘Y’, ‘Tb’, ‘Ho’), (‘Y’, ‘Tb’, ‘Er’), (‘Y’,‘Tb’, ‘Tm’), (‘Y’, ‘Tb’, ‘Yb’), (Y, ‘Tb’, ‘Lu’), (Y, Dy′, ‘Ho’), (‘Y’,Dy′, ‘Er’), (Y, ‘Dy’, ‘Tm’), (Y, ‘Dy’, ‘Yb’), (‘Y’, ‘Dy’, ‘Lu’), (‘Y’,‘Ho’, ‘Er’), (‘Y’, ‘Ho’, ‘Tm’), (‘Y’, ‘Ho’, ‘Yb’), (‘Y’, ‘Ho’, ‘Lu’),(‘Y’, ‘Er’, ‘Tm’), (‘Y’, ‘Er’, ‘Y b’), (‘Y’, ‘Er’, ‘Lu’), (‘Y’, ‘Tm’,‘Yb’), (‘Y’, ‘Tm’, ‘Lu’), (‘Y’, ‘Yb’, ‘Lu’), (‘La’, ‘Ce’, ‘Pr’), (‘La’,‘Ce’, ‘Nd’), (‘La’, ‘Ce’, ‘Sm’), (‘La’, ‘Ce’, ‘Eu’), (‘La’, ‘Ce’, ‘Gd’),(‘La’, ‘Ce’, ‘Tb’), (‘La’, ‘Ce’, ‘Dy’), (‘La’, ‘Ce’, ‘ Ho’), (‘La’,‘Ce’, ‘Er’), (‘La’, ‘Ce’, ‘Tm’), (‘La’, ‘Ce’, ‘Yb’), (‘La’, ‘Ce’, ‘Lu’),(‘La’, ‘Pr’, ‘Nd’), (‘La’, ‘Pr’, ‘Sm’), (‘La’, ‘Pr’, ‘Eu’), (‘La’, ‘Pr’,‘Gd’), (‘La’, ‘Pr’, ‘Tb’), (‘La’, ‘Pr’, ‘Dy’), (‘La’, ‘Pr’, ‘Ho’),(‘La’, ‘Pr’, ‘ Er’), (‘La’, ‘Pr’, ‘Tm’), (‘La’, ‘Pr’, ‘Yb’), (‘La’,‘Pr’, ‘Lu’), (‘La’, ‘Nd’, ‘Sm’), (‘La’, ‘Nd’, ‘Eu’), (‘La’, ‘Nd’, ‘Gd’),(‘La’, ‘Nd’, ‘Tb’), (‘La’, ‘Nd’, ‘Dy’), (‘La’, ‘Nd’, ‘Ho’), (‘La’, ‘Nd’,‘Er’), (‘La’, ‘Nd’, ‘Tm’), (‘La’, ‘ Nd’, ‘Yb’), (‘La’, ‘Nd’, ‘Lu’),(‘La’, ‘Sm’, ‘Eu’), (‘La’, ‘Sm’, ‘Gd’), (‘La’, ‘Sm’, ‘Tb’), (‘La’, ‘Sm’,‘Dy’), (′ La′, ‘Sm’, ‘Ho’), (‘La’, ‘Sm’, ‘Er’), (‘La’, ‘Sm’, ‘Tm’),(‘La’, ‘Sm’, ‘Yb’), (‘La’, ‘Sm’, ‘Lu’), (‘La’, ‘Eu’, ‘G d’), (‘La’,‘Eu’, ‘Tb’), (‘La’, ‘Eu’, ‘Dy’), (‘La’, ‘Eu’, ‘Ho’), (‘La’, ‘Eu’, ‘Er’),(‘La’, ‘Eu’, ‘Tm’), (‘La’, ‘Eu’, ‘ Yb’), (‘La’, ‘Eu’, ‘Lu’), (‘La’,‘Gd’, ‘Tb’), (‘La’, ‘Gd’, ‘Dy’), (‘La’, ‘Gd’, ‘Ho’), (‘La’, ‘Gd’, ‘Er’),(‘La’, ‘G d’, ‘Tm’), (‘La’, ‘Gd’, ‘Yb’), (‘La’, ‘Gd’, ‘Lu’), (‘La’,‘Tb’, ‘Dy’), (‘La’, ‘Tb’, ‘Ho’), (‘La’, ‘Tb’, ‘Er’), (‘La’, ‘Tb’, ‘Tm’),(‘La’, ‘Tb’, ‘Yb’), (‘La’, ‘Tb’, ‘Lu’), (‘La’, ‘Dy’, ‘Ho’), (‘La’, ‘Dy’,‘Er’), (‘La’, ‘Dy’, ‘Tm’), (′L a′, ‘Dy’, ‘Yb’), (‘La’, ‘Dy’, ‘Lu’),(‘La’, ‘Ho’, ‘Er’), (‘La’, ‘Ho’, ‘Tm’), (‘La’, ‘Ho’, ‘Yb’), (‘La’, ‘Ho’,‘Lu’), (‘La’, ‘Er’, ‘Tm’), (‘La’, ‘Er’, ‘Yb’), (‘La’, ‘Er’, ‘Lu’),(‘La’, ‘Tm’, ‘Yb’), (‘La’, ‘Tm’, ‘Lu’), (‘La’, ‘Yb’, ‘Lu’), (‘Ce’, ‘Pr’,‘Nd’), (‘Ce’, ‘Pr’, ‘Sm’), (‘Ce’, ‘Pr’, ‘Eu’), (‘Ce’, ‘Pr’, ‘Gd’),(‘Ce’, ‘Pr’, ‘Tb’), (‘Ce’, ‘Pr’, ‘Dy’), (‘Ce’, ‘Pr’, ‘Ho’), (‘Ce’, ‘Pr’,‘Er’), (‘Ce’, ‘Pr’, ‘Tm’), (‘Ce’, ‘Pr’, ‘Yb’), (‘Ce’, ‘Pr’, ‘Lu’),(‘Ce’, ‘Nd’, ‘Sm’), (‘Ce’, ‘Nd’, ‘Eu’), (‘Ce’, ‘Nd’, ‘Gd’), (‘Ce’, ‘Nd’,‘Tb’), (‘Ce’, ‘Nd’, ‘Dy’), (‘Ce’, ‘Nd’, ‘Ho’), (‘Ce’, ‘Nd’, ‘Er’),(‘Ce’, ‘Nd’, ‘Tm’), (‘Ce’, ‘Nd’, ‘Yb’), (‘Ce’, ‘Nd’, ‘Lu’), (‘Ce’, ‘Sm’,‘Eu’), (‘Ce’, ‘Sm’, ‘Gd’), (‘Ce’, ‘ Sm’, ‘Tb’), (‘Ce’, ‘Sm’, ‘Dy’),(‘Ce’, ‘Sm’, ‘Ho’), (‘Ce’, ‘Sm’, ‘Er’), (‘Ce’, ‘Sm’, ‘Tm’), (‘Ce’, ‘Sm’,‘Yb’), (‘Ce’, ‘Sm’, ‘Lu’), (‘Ce’, ‘Eu’, ‘Gd’), (‘Ce’, ‘Eu’, ‘Tb’),(‘Ce’, ‘Eu’, ‘Dy’), (‘Ce’, ‘Eu’, ‘Ho’), (‘Ce’, ‘Eu’, ‘Er’), (‘Ce’, ‘Eu’,‘Tm’), (‘Ce’, ‘Eu’, ‘Yb’), (‘Ce’, ‘Eu’, ‘Lu’), (‘Ce’, ‘Gd’, ‘Tb’),(‘Ce’, ‘Gd’, ‘Dy’), (‘Ce’, ‘Gd’, ‘Ho’), (‘Ce’, ‘Gd’, ‘Er’), (‘Ce’, ‘Gd’,‘Tm’), (‘Ce’, ‘Gd’, ‘Yb’), (‘Ce’, ‘Gd’, ‘Lu’), (‘Ce’, ‘Tb’, ‘Dy’),(‘Ce’, ‘ Tb’, ‘Ho’), (‘Ce’, ‘Tb’, ‘Er’), (‘Ce’, ‘Tb’, ‘Tm’), (‘Ce’,‘Tb’, ‘Yb’), (‘Ce’, ‘Tb’, ‘Lu’), (‘Ce’, ‘Dy’, ‘Ho’), (′C e′, ‘Dy’,‘Er’), (‘Ce’, ‘Dy’, ‘Tm’), (‘Ce’, ‘Dy’, ‘Yb’), (‘Ce’, ‘Dy’, ‘Lu’),(‘Ce’, ‘Ho’, ‘Er’), (‘Ce’, ‘Ho’, ‘Tm’), (‘Ce’, ‘Ho’, ‘Yb’), (‘Ce’, ‘Ho’,‘Lu’), (‘Ce’, ‘Er’, ‘Tm’), (‘Ce’, ‘Er’, ‘Yb’), (‘Ce’, ‘Er’, ‘Lu’),(‘Ce’, ‘Tm’, ‘ Yb’), (‘Ce’, ‘Tm’, ‘Lu’), (‘Ce’, ‘Yb’, ‘Lu’), (‘Pr’,‘Nd’, ‘Sm’), (‘Pr’, ‘Nd’, ‘Eu’), (‘Pr’, ‘Nd’, ‘Gd’), (‘Pr’, ‘N d’,‘Tb’), (‘Pr’, ‘Nd’, ‘Dy’), (‘Pr’, ‘Nd’, ‘Ho’), (‘Pr’, ‘Nd’, ‘Er’),(‘Pr’, ‘Nd’, ‘Tm’), (‘Pr’, ‘Nd’, ‘Yb’), (‘Pr’, ‘ Nd’, ‘Lu’), (‘Pr’,‘Sm’, ‘Eu’), (‘Pr’, ‘Sm’, ‘Gd’), (‘Pr’, ‘Sm’, ‘Tb’), (‘Pr’, ‘Sm’, ‘Dy’),(‘Pr’, ‘Sm’, ‘Ho’), (‘Pr’, ‘Sm’, ‘Er’), (‘Pr’, ‘Sm’, ‘Tm’), (‘Pr’, ‘Sm’,‘Yb’), (‘Pr’, ‘Sm’, ‘Lu’), (‘Pr’, ‘Eu’, ‘Gd’), (‘Pr’, ‘Eu’, ‘Tb’), (′Pr′, ‘Eu’, ‘Dy’), (‘Pr’, ‘Eu’, ‘Ho’), (‘Pr’, ‘Eu’, ‘Er’), (‘Pr’, ‘Eu’,‘Tm’), (‘Pr’, ‘Eu’, ‘Yb’), (‘Pr’, ‘Eu’, ‘Lu’), (′ Pr′, ‘Gd’, ‘Tb’),(‘Pr’, ‘Gd’, ‘Dy’), (‘Pr’, ‘Gd’, ‘Ho’), (‘Pr’, ‘Gd’, ‘Er’), (‘Pr’, ‘Gd’,‘Tm’), (‘Pr’, ‘Gd’, ‘Yb’), (‘Pr’, ‘Gd’, ‘Lu’), (‘Pr’, ‘Tb’, ‘Dy’),(‘Pr’, ‘Tb’, ‘Ho’), (‘Pr’, ‘Tb’, ‘Er’), (‘Pr’, ‘Tb’, ‘Tm’), (‘Pr’, ‘Tb’,‘Yb’), (‘Pr’, ‘Tb’, ‘Lu’), (‘Pr’, ‘Dy’, ‘Ho’), (‘Pr’, ‘Dy’, ‘Er’),(‘Pr’, ‘Dy’, ‘Tm’), (‘Pr’, ‘Dy’, ‘Yb’), (‘Pr’, ‘Dy’, ‘Lu’), (‘Pr’, ‘Ho’,‘Er’), (‘Pr’, ‘Ho’, ‘Tm’), (‘Pr’, ‘Ho’, ‘Yb’), (‘Pr’, ‘Ho’, ‘Lu’),(‘Pr’, ‘Er’, ‘Tm’), (‘Pr’, ‘Er’, ‘Yb’), (‘Pr’, ‘Er’, ‘Lu’), (‘Pr’, ‘Tm’,‘Yb’), (‘Pr’, ‘Tm’, ‘Lu’), (‘Pr’, ‘Yb’, ‘Lu’), (‘Nd’, ‘Sm’, ‘Eu’),(‘Nd’, ‘Sm’, ‘Gd’), (‘Nd’, ‘Sm’, ‘Tb’), (‘Nd’, ‘Sm’, ‘Dy’), (‘Nd’, ‘Sm’,‘Ho’), (‘Nd’, ‘Sm’, ‘Er’), (‘Nd’, ‘Sm’, ‘Tm’), (‘Nd’, ‘ Sm’, ‘Yb’),(‘Nd’, ‘Sm’, ‘Lu’), (‘Nd’, ‘Eu’, ‘Gd’), (‘Nd’, ‘Eu’, ‘Tb’), (‘Nd’, ‘Eu’,‘Dy’), (‘Nd’, ‘Eu’, ‘Ho’), (′ Nd′, ‘Eu’, ‘Er’), (‘Nd’, ‘Eu’, ‘Tm’),(‘Nd’, ‘Eu’, ‘Yb’), (‘Nd’, ‘Eu’, ‘Lu’), (‘Nd’, ‘Gd’, ‘Tb’), (‘Nd’, ‘Gd’,‘D y’), (‘Nd’, ‘Gd’, ‘Ho’), (‘Nd’, ‘Gd’, ‘Er’), (‘Nd’, ‘Gd’, ‘Tm’),(‘Nd’, ‘Gd’, ‘Yb’), (‘Nd’, ‘Gd’, ‘Lu’), (‘Nd’, ‘ Tb’, ‘Dy’), (‘Nd’,‘Tb’, ‘Ho’), (‘Nd’, ‘Tb’, ‘Er’), (‘Nd’, ‘Tb’, ‘Tm’), (‘Nd’, ‘Tb’, ‘Yb’),(‘Nd’, ‘Tb’, ‘Lu’), (′ Nd′, ‘Dy’, ‘Ho’), (‘Nd’, ‘Dy’, ‘Er’), (‘Nd’,‘Dy’, ‘Tm’), (‘Nd’, ‘Dy’, ‘Yb’), (‘Nd’, ‘Dy’, ‘Lu’), (‘Nd’, ‘Ho’, ‘Er’), (‘Nd’, ‘Ho’, ‘Tm’), (‘Nd’, ‘Ho’, ‘Yb’), (‘Nd’, ‘Ho’, ‘Lu’), (‘Nd’,‘Er’, ‘Tm’), (‘Nd’, ‘Er’, ‘Yb’), (‘Nd’, ‘ Er’, ‘Lu’), (‘Nd’, ‘Tm’,‘Yb’), (‘Nd’, ‘Tm’, ‘Lu’), (‘Nd’, ‘Yb’, ‘Lu’), (‘Sm’, ‘Eu’, ‘Gd’),(‘Sm’, ‘Eu’, ‘Tb’), (‘Sm’, ‘Eu’, ‘Dy’), (‘Sm’, ‘Eu’, ‘Ho’), (‘Sm’, ‘Eu’,‘Er’), (‘Sm’, ‘Eu’, ‘Tm’), (‘Sm’, ‘Eu’, ‘Yb’), (‘Sm’, ‘Eu’, ‘ Lu’),(‘Sm’, ‘Gd’, ‘Tb’), (‘Sm’, ‘Gd’, ‘Dy’), (‘Sm’, ‘Gd’, ‘Ho’), (‘Sm’, ‘Gd’,‘Er’), (‘Sm’, ‘Gd’, ‘Tm’), (′Sm ‘Gd’, ‘Yb’), (‘Sm’, ‘Gd’, ‘Lu’), (‘Sm’,‘Tb’, ‘Dy’), (‘Sm’, ‘Tb’, ‘Ho’), (‘Sm’, ‘Tb’, ‘Er’), (‘Sm’, ‘Tb’, ‘Tm’),(‘Sm’, ‘Tb’, ‘Yb’), (‘Sm’, ‘Tb’, ‘Lu’), (‘Sm’, ‘Dy’, ‘Ho’), (‘Sm’, ‘Dy’,‘Er’), (‘Sm’, ‘Dy’, ‘Tm’), (‘Sm’, ‘D y’, ‘Yb’), (‘Sm’, ‘Dy’, ‘Lu’),(‘Sm’, ‘Ho’, ‘Er’), (‘Sm’, ‘Ho’, ‘Tm’), (‘Sm’, ‘Ho’, ‘Yb’), (‘Sm’, ‘Ho’,‘Lu’), (′ Sm′, ‘Er’, ‘Tm’), (‘Sm’, ‘Er’, ‘Yb’), (‘Sm’, ‘Er’, ‘Lu’),(‘Sm’, ‘Tm’, ‘Yb’), (‘Sm’, ‘Tm’, ‘Lu’), (‘Sm’, ‘Yb’, ‘ Lu’), (‘Eu’,‘Gd’, ‘Tb’), (‘Eu’, ‘Gd’, ‘Dy’), (‘Eu’, ‘Gd’, ‘Ho’), (‘Eu’, ‘Gd’, ‘Er’),(‘Eu’, ‘Gd’, ‘Tm’), (‘Eu’, ‘ Gd’, ‘Yb’), (‘Eu’, ‘Gd’, ‘Lu’), (‘Eu’,‘Tb’, ‘Dy’), (‘Eu’, ‘Tb’, ‘Ho’), (‘Eu’, ‘Tb’, ‘Er’), (‘Eu’, ‘Tb’, ‘Tm’),(′E u′, ‘Tb’, ‘Yb’), (‘Eu’, ‘Tb’, ‘Lu’), (‘Eu’, ‘Dy’, ‘Ho’), (‘Eu’,‘Dy’, ‘Er’), (‘Eu’, ‘Dy’, ‘Tm’), (‘Eu’, ‘Dy’, ‘Yb’), (‘Eu’, ‘Dy’, ‘Lu’),(‘Eu’, ‘Ho’, ‘Er’), (‘Eu’, ‘Ho’, ‘Tm’), (‘Eu’, ‘Ho’, ‘Yb’), (‘Eu’, ‘Ho’,‘Lu’), (‘Eu’, ‘Er’, ‘T m’), (‘Eu’, ‘Er’, ‘Yb’), (‘Eu’, ‘Er’, ‘Lu’),(‘Eu’, ‘Tm’, ‘Yb’), (‘Eu’, ‘Tm’, ‘Lu’), (‘Eu’, ‘Yb’, ‘Lu’), (‘Gd’, ‘Tb’,‘Dy’), (‘Gd’, ‘Tb’, ‘Ho’), (‘Gd’, ‘Tb’, ‘Er’), (‘Gd’, ‘Tb’, ‘Tm’),(‘Gd’, ‘Tb’, ‘Yb’), (‘Gd’, ‘Tb’, ‘Lu’), (‘Gd’, ‘Dy’, ‘Ho’), (‘Gd’, ‘Dy’,‘Er’), (‘Gd’, ‘Dy’, ‘Tm’), (‘Gd’, ‘Dy’, ‘Yb’), (‘Gd’, ‘Dy’, ‘Lu’),(‘Gd’, ‘Ho’, ‘Er’), (‘Gd’, ‘Ho’, ‘Tm’), (‘Gd’, ‘Ho’, ‘Yb’), (‘Gd’, ‘Ho’,‘Lu’), (‘Gd’, ‘Er’, ‘Tm’), (‘Gd’, ‘Er’, ‘Yb’), (‘Gd’, ‘Er’, ‘ Lu’),(‘Gd’, ‘Tm’, ‘Yb’), (‘Gd’, ‘Tm’, ‘Lu’), (‘Gd’, ‘Yb’, ‘Lu’), (‘Tb’, ‘Dy’,‘Ho’), (‘Tb’, ‘Dy’, ‘Er’), (‘Tb’, ‘ Dy’, ‘Tm’), (‘Tb’, ‘Dy’, ‘Yb’),(‘Tb’, ‘Dy’, ‘Lu’), (‘Tb’, ‘Ho’, ‘Er’), (‘Tb’, ‘Ho’, ‘Tm’), (‘Tb’, ‘Ho’,‘Yb’), (′ Tb′, ‘Ho’, ‘Lu’), (‘Tb’, ‘Er’, ‘Tm’), (‘Tb’, ‘Er’, ‘Yb’),(‘Tb’, ‘Er’, ‘Lu’), (‘Tb’, ‘Tm’, ‘Yb’), (‘Tb’, ‘Tm’, ‘Lu’), (‘Tb’, ‘Yb’,‘Lu’), (‘Dy’, ‘Ho’, ‘Er’), (‘Dy’, ‘Ho’, ‘Tm’), (‘Dy’, ‘Ho’, ‘Yb’),(‘Dy’, ‘Ho’, ‘Lu’), (‘Dy’, ‘Er’, ‘Tm’), (‘Dy’, ‘Er’, ‘Yb’), (‘Dy’, ‘Er’,‘Lu’), (‘Dy’, ‘Tm’, ‘Yb’), (‘Dy’, ‘Tm’, ‘Lu’), (‘Dy’, ‘Yb’, ‘Lu’),(‘Ho’, ‘Er’, ‘Tm’), (‘Ho’, ‘Er’, ‘Yb’), (‘Ho’, ‘Er’, ‘Lu’), (‘Ho’, ‘Tm’,‘Yb’), (‘Ho’, ‘Tm’, ‘Lu’), (‘Ho’, ‘Yb’, ‘Lu’), (′ Er′, ‘Tm’, ‘Yb’),(‘Er’, ‘Tm’, ‘Lu’), (‘Er’, ‘Yb’, ‘Lu’), (‘Tm’, ‘Yb’, ‘Lu’)

-   -   3. Example Verification methodology 3: this methodology expands        on Verification methodology 2 by the use of naturally occurring        microbial cation exchangers, including bacteria and fungi.        -   i. As above but right before step vii (unsealing samples in            laboratory), add the following steps:        -   ii. Lyse the microbes. For example, autoclave each            sample-containing beaker for steam sterilization at 250° F.            at 15 psi for 15 minutes. This may lyse any microbes and            cause an active release of bio-adsorbed metals.        -   iii. Let sample-containing beakers cool to room temperature            before proceeding.        -   iv. The filter in step (ix) may now act to additionally            remove any larger aggregates of microbial cell suspensions            that did not lyse in the autoclaving process.        -   v. The added EDTA in step (x) in Verification Methodology 2            may act to additionally chelate any REEs that are complexed            to the cell wall or organic molecules of the microbial            genetic material (as a result of cell lysis).        -   vi. In conjunction, verification methodology 3 provides a            protocol to achieve full recovery of REEs that were retained            in the top 10 cm of soil due to the following biogeochemical            processes:            -   1. Mineral surface interface complexation.            -   2. The cation exchange capacity inherent to soils due to                organic matter.            -   3. Microbially mediated surface biosorption and/or                active biological absorption pathways including but not                limited to REE-aqueous complexation with internal                genetic material of individual microbial cells found in                natural soils and sediments.                B. Example systems and methods to enhance the agronomic                performance and ecosystem co-benefits:    -   In these non-limiting embodiments, product formulations are        detailed that improve the soil quality for improved agronomic        applications as well as embodiments that specify improvements on        crop health and an ability for plants to protect themselves from        pathogens.    -   Addition of macronutrients that enhance the agronomic benefits        of the applied mineral. Example: addition of elemental 1-5% by        weight formulation K or Ca or inexpensive REE such as La to        improve use cases of pulverized Mg₂SiO₄ by improving the        nutrient balance of the soil to match the needs of actively        growing plants. Application rates for K and Ca may range from        50-100 ppm and lanthanide addition rates may range from 1-50        ppm.    -   The 1-5% by volume addition of slow-release acidifiers in the        form of elements that maintain the acidity of the soil despite        the tendency of the applied mineral to reduce soil acidity in        the weathering process. Example: addition of minerals with Al        (such as gibbsite, Al(OH)₃) or S (such as gypsum, CaSO₄)-2H₂O),        in a rate of 1-10% by weight of total formulation, which        contribute to maintaining soil acidity, and thus maintain high        rates of weathering, which counteracts the tendency of        forsterite weathering to increase alkalinity, which slows rates        of weathering.

The incorporation of additional micronutrients such as zinc (1-5% byweight compared to mineral amendment) to reduce ecosystem losses ofphosphorus and downstream ecosystem impacts of phosphate-basedfertilizers, such as eutrophication. Phosphates are immobilized viaprecipitation reactions with the additional zinc metal, so zinc additionslows the transport and reduces mobility of phosphate in the subsoil. Asa consequence, phosphate-based fertilizer applications can be performedmore safely and with higher confidence that downstream waterways willnot form algal blooms and lead to anoxic, uninhabitable waterways foraquatic life. The addition of 1-32 tons/hectare of mineral inconjunction with 1-5% by weight zinc in fields with heavy phosphateapplication reduces the likelihood of significant phosphate leaching.This also increases nutrient use efficiency, which reduces fertilizercost to farmers and reduces negative environmental impacts to society.Specifically, phosphate-zinc precipitates become slow-releasing overtime due to their immobilization in the soil as a solid phase. Thisprovides longer timeframes for plants to access the phosphateapplication, reducing the frequency of fertilization.

-   -   The method of mixing of different mineral adjuvants to the        primary mineral used for carbon removal can be varied, with        impacts on performance. In some embodiments, the admixture        (e.g., of gypsum and forsterite) could be completely emulsified.        In some embodiments, a nutrient, such as urea, could represent a        core that is subsequently coated with a shell of silicate        mineral used for carbon removal. In some embodiments the        silicate mineral could be the core, and the nutrient, such as        urea, could be the coating. In each of these cases, the        embodiment can be optimized so as to reduce environmental losses        of the nutrient, and increase availability to the plant, while        providing the acidity necessary for the silicate mineral        weathering. In some embodiments, this may represent a “slow        release” nutrient that does not mineralize too quickly and is        synchronized more favorably with plant demand.    -   Comminution, or pulverization, may help reproduce the expected        kinetics of alkalizers such as pulverized limestone or dolomite,        known as aglime. Aglime has well-characterized kinetics        originating in the particle size distribution of the product,        which are in some jurisdictions legally regulated to meet        certain requirements. A representative cross section of        dissolution kinetics based on actual aglime mesh size        observations is depicted in FIG. 2 .    -   It naturally follows that agronomic performance expectations for        silicate minerals used for ERW should follow similar reaction        kinetics, particularly in the first year after application,        while also meeting the goals for CDR. Thus, an ideal particle        size distribution may be tuned to meet this goal. The following        shows the kinetics for a particle size distribution of a        pulverized silicate with median particle size of 80-100 um, such        as 90 um, which closely approximates the alkalinity release        dynamics for aglime.    -   The small particle sizes produced by pulverization as above are        valuable for performance as an aglime substitute and CDR        mechanism, but introduce their own set of potential challenges.        One potential challenge is that small particle sizes may fall        within regulated categories such as PM10 or PM2.5; another        potential challenge is the product may not be easily transferred        from one vessel to another (so-called “flowability”) because it        has a tendency to settle into a compact mass; the product may        also not be applied uniformly by extant farm equipment;        potentially low uniformity application impacts the agronomic        performance as well as the sampling density needed for carbon        removal verification. In one potential formulation, the product        is pelletized after pulverization, using a common binder such as        lignosulfate in a 3-5% ratio by weight of the total formulation.        The amount of binder may be optimized to improve performance        characteristics during transport and application, but minimize        the water and energy needed to dissolve the pellet once it is in        the field. Pelletizing the product, into a size range of, for        example, 0.5 mm to 3 mm increases the flowability; reduces the        prevalence of dust, including the risk of asbestos exposure;        enables the use of extant crop input application equipment; and        improves the evenness of field distribution, which has the        agronomic and carbon verification benefits identified.        Pelletizing in this size range also ensures that an adequate        number of pellets fall onto any land area which may subsequently        be sampled for analysis.    -   A desiccated but living biological compound, such as a        mycorrhizal inoculum, may be used to coat the agglomerated and        dried pellets, so as to enhance the dissolution of the pellet        itself after application. This may be distinct from a fungal        adjuvant to the pellet to increase dissolution of the silicate        minerals themselves.    -   To accommodate a range of soil types from acidic to neutral pH,        the product may be differentiated into different particle size        distributions so as to achieve a consistent rate of weathering        across different soil types. For example a consistent rate of        weathering may be that the entire applied amount of mineral        product weathers in 9 months, allowing a subsequent application        every year. To achieve this, an acidic soil may have pellets        that contain particles with a modal size on the order of 100 um,        while a neutral soil may have pellets that contain particles        with a modal size on the order of 10 um. The negative        consequence of applying a fine particle in an acidic soil may be        that it dissolves too quickly, swinging the pH of the soil too        quickly, and altering the nutrient availability severely in the        growing plant. The negative consequence of applying a coarse        particle in a neutral soil may be that the particle weathers too        slowly or is functionally inert, which does not remove carbon as        intended, and may subject the farmer or the vendor to risk of        clawbacks for payments received for carbon removal.        C. Example systems and methods to control or enhance the        performance as a securitized carbon removal method:    -   In these non-limiting embodiments, product formulations are        described that improve the marketability of pulverized and        optionally pelletized mineral amendments as a verifiable carbon        removal method, irrespective of their impact on agronomic        performance. Such changes in marketability increase the value of        the carbon product, for example for detecting and preventing        fraud, or increase the dissolution rate (mass per area per year)        to accelerate the timing of reapplication, which raises the        value in a discounted cash flow analysis. The general scheme is        depicted in FIG. 4 .    -   The addition of low cost and inert trace elements, for example        1-5 ppm neodymium or lanthanum, to a mineral amendment, beyond        the natural abundances present in the ore body, improves        performance in Verification methods 2 and 3. Specifically, the        additional elements reduce measurement uncertainty owing to low        natural abundance calculations as well as uncertainty owing to        variability from the source mineral. Reduction of these        uncertainties improves the detectability of true positives and        true negatives. This principle could be applied to many        potential soil amendments and fertilizers to verify provenance        as pertains to legal contracts.    -   The addition of binders to create aggregations of the ground        mineral that could reduce drift, which improves distribution        uniformity; reduces potential respiratory health impacts; and/or        improves applicability of the applied mineral using commercially        available equipment. For example, a biofilm mediating organic        compound, such as alginate or chitosan, could be mixed with a        pulverized mineral with a particle size of approximately 100        microns to achieve an aggregated pellet size of 1 millimeter.        This would maintain the advantages of fine particle size,        particularly the high specific surface area (m²/g) that mediates        dissolution rate, while offsetting potential limitations        outlined above. An example methodology for aggregating ground        minerals could include these steps:        -   Dissolve 10 grams of alginate powder into solution to create            a 0.5 M-1.0 M alginic acid solution.        -   Heat solution to 80-100° F. to dissolve alginate powder            fully if necessary.        -   Spray aerosol sized particles of dissolved alginate powder            onto the ground rock material in order to create a            biofilm-coated rock material. Apply a consistent spray of            100 mL for each 100 gram of rock (1:1 v/w).        -   The resulting aggregates may have increased bulk density as            well as higher aggregating properties due to van der Waals            force attraction between the individual rock particles.    -   Dissolution rates can be modified by non-living organic        molecules applied concurrently with the mineral amendment. For        example, protonated microbes such as acid-treated Arthrobacter        nicotianae can constitute an initial release of protons to        attack the ground rock materials' crystal structure. This method        of acidification for optimal accelerated weathering of rock        material may occur even during winter/cold climate. Because        biologically mediated processes can have sharp thresholds, e.g.,        under freezing conditions, such a chemically controlled process        would exhibit greater ability to expedite rock breakdown over a        range of environmental conditions. To implement this        methodology, we outline the following steps:        -   1) De-frost an aliquot of A. nicotianae in 5 mL of Luria            broth for about 24 hours at exactly 30° C.        -   2) Sub-culture into a larger volume of Luria broth at a 1:50            dilution and allow growth for another 24 hours at 30° C.        -   3) During growth steps #1 and 2, cultures may be shaken, for            example at 200-220 RPM, for aeration.        -   4) After culture has been grown for 48 hours in total,            centrifuge the cell suspension.        -   5) Collect cell pellet and take a small amount and set it            aside. Weigh wet weight, dry 60° C. overnight, and weigh dry            amount to obtain wet: dry conversion.        -   6) Develop an OD₆₀₀ to wet weight to dry weight conversion            by taking 5 varying amounts of wet aliquots of wet biomass            and measuring optical density given its known wet mass. Wet            aliquots may have a mass of about 1, 4, 8, 12, 16 wet grams            and be suspended in 100 mL of DI water.        -   7) Wash cells in 0.1 M NaCl solution.            -   a. Centrifuge cells+growth media at 4000×g.            -   b. Carefully pour out LB broth.            -   c. Add 0.1 M NaCl solution.            -   d. Centrifuge cells+NaCl solution at 4000×g.            -   e. Carefully pour out NaCl solution.        -   8) Protonate the cells while wet.            -   a. Re-suspend the wet biomass in an acidic (e.g., pH                3.0) solution using a dilute hydrochloric acid solution                (0.001-0.01 M).            -   b. Centrifuge cells+HCl solution at 4000×g.            -   c. Carefully pour out solution.            -   d. Repeat steps (a)-(c) a total of three times.            -   e. Dry cells in an oven at 80-100° C. for 48 hours.            -   f. Grind cells using a mortar/pestle to achieve a powder                form.        -   9) Mix 5% w/w of dry cell powder with the ground rock            material.        -   10) When this particular rock material+cell powder mixture            is applied to agricultural soils, the irrigation or rainfall            will re-apply moisture and allow the pre-protonated, re-wet            biomass to release their protons. This will ultimately            induce a local acidity effect on the rock particle            (molecular) level to accelerate rock crystal lattice            breakdown. The cells act as biodegradable transporters that            are also benign to the natural soil environment.

What is claimed is:
 1. A method for verifying enhanced rock weatheringusing a mineral amendment comprising: measuring one or more immobiletrace elements in a mineral amendment, the immobile trace elementscomprising one or more rare earth elements, rare metals, or transitionmetals; and subsequently measuring the immobile trace elements in a soilsample after application of the mineral amendment to verify applicationof the mineral amendment.
 2. The method of claim 1, further comprisingthe step of determining the amount of the mineral amendment added to thesoil sample.
 3. The method of claim 1, further comprising the step ofdetermining the amount of measured divalent cations in the soil sample.4. The method of claim 3, further comprising the step of determining thefractional carbon dioxide removal by comparing the calculated amount ofapplied divalent cations to the measured divalent cations in the soilsample.
 5. The method of claim 1, wherein the immobile trace elementsand magnesium are measured using inductively coupled plasma spectroscopyor x-ray fluorescence spectroscopy.
 6. The method of claim 1, furthercomprising the step of measuring the immobile trace elements in the soilsample prior to addition of the mineral amendment.
 7. The method ofclaim 1, further comprising the step of enriching the abundance of theimmobile trace elements in the mineral amendment.
 8. The method of claim1, wherein: the rare earth elements comprise scandium (Sc), yttrium (Y),lanthanum (La), cerium (Ce), praseodymium (Pr), neodymium (Nd), samarium(Sm), europium (Eu), gadolinium (Gd), terbium (Tb), dysprosium (Dy),holmium (Ho), erbium (Er), thulium (Tm), ytterbium (Yb), and lutetium(Lu); the rare metals comprise beryllium (Be), cesium (Cs), gallium(Ga), germanium (Ge), hafnium (Hf), niobium (Nb), rubidium (Rb),tantalum (Ta), titanium (Ti), vanadium (V), and zirconium (Zr); and thetransition metals comprise nickel (Ni), chromium (Cr), and zinc (Zn). 9.The method of claim 1, wherein three immobile trace elements aremeasured.
 10. The method of claim 8, wherein the three immobile traceelements are selected from the list comprising: (‘Y’, ‘La’, ‘Ce’), (‘Y’,‘La’, ‘Pr’), (‘Y’, ‘La’, Nd′), (‘Y’, ‘La’, ‘Sm’), (‘Y’, ‘La’, ‘Eu’),(‘Y’, ‘La’, ‘Gd’), (‘Y’, ‘La’, ‘Tb’), (‘Y’, ‘La’, ‘Dy’), (‘Y’, ‘La’,‘Ho’), (‘Y’, ‘La’, ‘Er’), (‘Y’, ‘La’, ‘Tm’), (‘Y’, ‘La’, ‘Yb’), (‘Y’,‘La’, ‘Lu’), (‘Y’, ‘Ce’, ‘Pr’), (‘Y’, ‘Ce’, ‘Nd’), (‘Y’, ‘Ce’, ‘Sm’),(‘Y’, ‘Ce’, ‘Eu’), ‘Ce’, ‘Gd’), (‘Y’, ‘Ce’, ‘Tb’), (‘Y’, ‘Ce’, ‘Dy’),(‘Y’, ‘Ce’, ‘Ho’), (‘Y’, ‘Ce’, ‘Er’), (‘Y’, ‘Ce’, ‘Tm’), (‘Y’, ‘Ce’,‘Yb’), (‘Y’, ‘Ce’, ‘Lu’), (‘Y’, ‘Pr’, ‘Nd’), (‘Y’, ‘Pr’, ‘Sm’), (‘Y’,‘Pr’, ‘Eu’), (‘Y’, ‘Pr’, ‘Gd’), (‘Y’, ‘Pr’, ‘Tb’), (‘Y’, ‘Pr’, ‘Dy’),(‘Y’, ‘Pr’, ‘Ho’), (‘Y’, ‘Pr’, ‘Er’), (‘Y’, ‘Pr’, ‘Tm’), (‘Y’, ‘Pr’,‘Yb’), (‘Y’, ‘Pr’, ‘Lu’), (‘Y’, ‘Nd’, ‘Sm’), (‘Y’, ‘Nd’, ‘Eu’), (‘Y’,‘Nd’, ‘Gd’), (‘Y’, ‘Nd’, ‘Tb’), (‘Y’, ‘Nd’, ‘Dy’), (‘Y’, ‘Nd’, ‘Ho’),(‘Y’, ‘Nd’, ‘Er’), (‘Y’, Nd′, ‘Tm’), (‘Y’, ‘Nd’, ‘Yb’), (‘Y’, ‘Nd’,‘Lu’), (‘Y’, ‘Sm’, ‘Eu’), (‘Y’, ‘Sm’, ‘Gd’), (‘Y’, ‘Sm’, ‘Tb’), (‘Y’,‘Sm’, ‘Dy’), (‘Y’, ‘Sm’, ‘Ho’), (‘Y’, ‘Sm’, ‘Er’), (‘Y’, ‘Sm’, ‘Tm’),(‘Y’, ‘Sm’, ‘Yb’), (‘Y’, ‘Sm’, ‘Lu’), (‘Y’, ‘Eu’, ‘Gd’), (‘Y’, ‘Eu’,‘Tb’), (‘Y’, ‘Eu’, ‘Dy’), (‘Y’, ‘Eu’, ‘Ho’), (‘Y’, ‘Eu’, ‘Er’), (‘Y’,‘Eu’, ‘Tm’), (‘Y’, ‘Eu’, ‘Yb’), (‘Y’, ‘Eu’, ‘Lu’), (‘Y’, ‘Gd’, ‘Tb’),(‘Y’, ‘Gd’, ‘Dy’), (‘Y’, ‘Gd’, ‘Ho’), (‘Y’, ‘Gd’, ‘Er’), (‘Y’, ‘Gd’,‘Tm’), (‘Y’, ‘Gd’, ‘Yb’), (‘Y’, ‘Gd’, ‘Lu’), (‘Y’, ‘Tb’, ‘Dy’), (‘Y’,‘Tb’, ‘Ho’), (‘Y’, ‘Tb’, ‘Er’), (‘Y’, ‘Tb’, ‘Tm’), (‘Y’, ‘Tb’, ‘Yb’),(‘Y’, ‘Tb’, ‘Lu’), (‘Y’, ‘Dy’, ‘Ho’), (‘Y’, ‘Dy’, ‘Er’), (‘Y’, ‘Dy’,‘Tm’), (‘Y’, ‘Dy’, ‘Yb’), (‘Y’, ‘Dy’, ‘Lu’), (‘Y’, ‘Ho’, ‘Er’), (‘Y’,‘Ho’, ‘Tm’), (‘Y’, ‘Ho’, ‘Yb’), (‘Y’, ‘Ho’, ‘Lu’), (‘Y’, ‘Er’, ‘Tm’),(‘Y’, ‘Er’, ‘Yb’), (‘Y’, ‘Er’, ‘Lu’), (‘Y’, ‘Tm’, ‘Yb’), (‘Y’, ‘Tm’,‘Lu’), (‘Y’, ‘Yb’, ‘Lu’), (‘La’, ‘Ce’, ‘Pr’), (‘La’, ‘Ce’, ‘Nd’), (‘La’,‘Ce’, ‘Sm’), (‘La’, ‘Ce’, ‘Eu’), (‘La’, ‘Ce’, ‘Gd’), (‘La’, ‘Ce’, ‘Tb’),(‘La’, ‘Ce’, ‘Dy’), (‘La’, ‘Ce’, ‘Ho’), (‘La’, ‘Ce’, ‘Er’), (‘La’, ‘Ce’,‘Tm’), (‘La’, ‘Ce’, ‘Yb’), (‘La’, ‘Ce’, ‘Lu’), (‘La’, ‘Pr’, ‘Nd’),(‘La’, ‘Pr’, ‘Sm’), (‘La’, ‘Pr’, ‘Eu’), (‘La’, ‘Pr’, ‘Gd’), (‘La’, ‘Pr’,‘Tb’), (‘La’, ‘Pr’, ‘Dy’), (‘La’, ‘Pr’, ‘Ho’), (‘La’, ‘Pr’, ‘Er’),(‘La’, ‘Pr’, ‘Tm’), (‘La’, ‘Pr’, ‘Yb’), (‘La’, ‘Pr’, ‘Lu’), (‘La’, ‘Nd’,‘Sm’), (‘La’, ‘Nd’, ‘Eu’), (‘La’, Nd′, ‘Gd’), (‘La’, ‘Nd’, ‘Tb’), (‘La’,‘Nd’, ‘Dy’), (‘La’, ‘Nd’, ‘Ho’), (‘La’, ‘Nd’, ‘Er’), (‘La’, ‘Nd’, ‘Tm’),(‘La’, ‘Nd’, ‘Yb’), (‘La’, ‘Nd’, ‘Lu’), (‘La’, ‘Sm’, ‘Eu’), (‘La’, ‘Sm’,‘Gd’), (‘La’, ‘Sm’, ‘Tb’), (‘La’, ‘Sm’, ‘Dy’), (‘La’, ‘Sm’, ‘Ho’),(‘La’, ‘Sm’, ‘Er’), (‘La’, ‘Sm’, ‘Tm’), (‘La’, ‘Sm’, ‘Yb’), (‘La’, ‘Sm’,‘Lu’), (‘La’, ‘Eu’, ‘Gd’), (‘La’, ‘Eu’, ‘Tb’), (‘La’, ‘Eu’, ‘Dy’),(‘La’, ‘Eu’, ‘Ho’), (‘La’, ‘Eu’, ‘Er’), (‘La’, ‘Eu’, ‘Tm’), (‘La’, ‘Eu’,‘Yb’), (‘La’, ‘Eu’, ‘Lu’), (‘La’, ‘Gd’, ‘Tb’), (‘La’, ‘Gd’, ‘Dy’),(‘La’, ‘Gd’, ‘Ho’), (‘La’, ‘Gd’, ‘Er’), (‘La’, ‘Gd’, ‘Tm’), (‘La’, ‘Gd’,‘Yb’), (‘La’, ‘Gd’, ‘Lu’), (‘La’, ‘Tb’, ‘Dy’), (‘La’, ‘Tb’, ‘Ho’),(‘La’, ‘Tb’, ‘Er’), (‘La’, ‘Tb’, ‘Tm’), (‘La’, ‘Tb’, ‘Yb’), (‘La’, ‘Tb’,‘Lu’), (‘La’, ‘Dy’, ‘Ho’), (‘La’, ‘Dy’, ‘Er’), (‘La’, ‘Dy’, ‘Tm’),(‘La’, ‘Dy’, ‘Yb’), (‘La’, ‘Dy’, ‘Lu’), (‘La’, ‘Ho’, ‘Er’), (‘La’, ‘Ho’,‘Tm’), (‘La’, ‘Ho’, ‘Yb’), (‘La’, ‘Ho’, ‘Lu’), (‘La’, ‘Er’, ‘Tm’),(‘La’, ‘Er’, ‘Yb’), (‘La’, ‘Er’, ‘Lu’), (‘La’, ‘Tm’, ‘Yb’), (‘La’, ‘Tm’,‘Lu’), (‘La’, ‘Yb’, ‘Lu’), (‘Ce’, ‘Pr’, ‘Nd’), (‘Ce’, ‘Pr’, ‘Sm’),(‘Ce’, ‘Pr’, ‘Eu’), (‘Ce’, ‘Pr’, ‘Gd’), (‘Ce’, ‘Pr’, ‘Tb’), (‘Ce’, ‘Pr’,‘Dy’), (‘Ce’, ‘Pr’, ‘Ho’), (‘Ce’, ‘Pr’, ‘Er’), (‘Ce’, ‘Pr’, ‘Tm’),(‘Ce’, ‘Pr’, ‘Yb’), (‘Ce’, ‘Pr’, ‘Lu’), (‘Ce’, ‘Nd’, ‘Sm’), (‘Ce’, ‘Nd’,‘Eu’), (‘Ce’, ‘Nd’, ‘Gd’), (‘Ce’, ‘Nd’, ‘Tb’), (‘Ce’, ‘Nd’, ‘Dy’),(‘Ce’, ‘Nd’, ‘Ho’), (‘Ce’, ‘Nd’, ‘Er’), (‘Ce’, ‘Nd’, ‘Tm’), (‘Ce’, ‘Nd’,‘Yb’), (‘Ce’, ‘Nd’, ‘Lu’), (‘Ce’, ‘Sm’, ‘Eu’), (‘Ce’, ‘Sm’, ‘Gd’),(‘Ce’, ‘Sm’, ‘Tb’), (‘Ce’, ‘Sm’, ‘Dy’), (‘Ce’, ‘Sm’, ‘Ho’), (‘Ce’, ‘Sm’,‘Er’), (‘Ce’, ‘Sm’, ‘Tm’), (‘Ce’, ‘Sm’, ‘Yb’), (‘Ce’, ‘Sm’, ‘Lu’),(‘Ce’, ‘Eu’, ‘Gd’), (‘Ce’, ‘Eu’, ‘Tb’), (‘Ce’, ‘Eu’, ‘Dy’), (‘Ce’, ‘Eu’,‘Ho’), (‘Ce’, ‘Eu’, ‘Er’), (‘Ce’, ‘Eu’, ‘Tm’), (‘Ce’, ‘Eu’, ‘Yb’),(‘Ce’, ‘Eu’, ‘Lu’), (‘Ce’, ‘Gd’, ‘Tb’), (‘Ce’, ‘Gd’, ‘Dy’), (‘Ce’, ‘Gd’,‘Ho’), (‘Ce’, ‘Gd’, ‘Er’), (‘Ce’, ‘Gd’, ‘Tm’), (‘Ce’, ‘Gd’, ‘Yb’),(‘Ce’, ‘Gd’, ‘Lu’), (‘Ce’, ‘Tb’, ‘Dy’), (‘Ce’, ‘Tb’, ‘Ho’), (‘Ce’, ‘Tb’,‘Er’), (‘Ce’, ‘Tb’, ‘Tm’), (‘Ce’, ‘Tb’, ‘Yb’), (‘Ce’, ‘Tb’, ‘Lu’),(‘Ce’, ‘Dy’, ‘Ho’), (‘Ce’, ‘Dy’, ‘Er’), (‘Ce’, ‘Dy’, ‘Tm’), (‘Ce’, ‘Dy’,‘Yb’), (‘Ce’, ‘Dy’, ‘Lu’), (‘Ce’, ‘Ho’, ‘Er’), (‘Ce’, ‘Ho’, ‘Tm’),(‘Ce’, ‘Ho’, ‘Yb’), (‘Ce’, ‘Ho’, ‘Lu’), (‘Ce’, ‘Er’, ‘Tm’), (‘Ce’, ‘Er’,‘Yb’), (‘Ce’, ‘Er’, ‘Lu’), (‘Ce’, ‘Tm’, ‘Yb’), (‘Ce’, ‘Tm’, ‘Lu’),(‘Ce’, ‘Yb’, ‘Lu’), (‘Pr’, ‘Nd’, ‘Sm’), (‘Pr’, ‘Nd’, ‘Eu’), (‘Pr’, ‘Nd’,‘Gd’), (‘Pr’, ‘Nd’, ‘Tb’), (‘Pr’, ‘Nd’, ‘Dy’), (‘Pr’, ‘Nd’, ‘Ho’),(‘Pr’, ‘Nd’, ‘Er’), (‘Pr’, ‘Nd’, ‘Tm’), (‘Pr’, ‘Nd’, ‘Yb’), (‘Pr’, ‘Nd’,‘Lu’), (‘Pr’, ‘Sm’, ‘Eu’), (‘Pr’, ‘Sm’, ‘Gd’), (‘Pr’, ‘Sm’, ‘Tb’),(‘Pr’, ‘Sm’, ‘Dy’), (‘Pr’, ‘Sm’, ‘Ho’), (‘Pr’, ‘Sm’, ‘Er’), (‘Pr’, ‘Sm’,‘Tm’), (‘Pr’, ‘Sm’, ‘Yb’), (‘Pr’, ‘Sm’, ‘Lu’), (‘Pr’, ‘Eu’, ‘Gd’),(‘Pr’, ‘Eu’, ‘Tb’), (‘Pr’, ‘Eu’, ‘Dy’), (‘Pr’, ‘Eu’, ‘Ho’), (‘Pr’, ‘Eu’,‘Er’), (‘Pr’, ‘Eu’, ‘Tm’), (‘Pr’, ‘Eu’, ‘Yb’), (‘Pr’, ‘Eu’, ‘Lu’),(‘Pr’, ‘Gd’, ‘Tb’), (‘Pr’, ‘Gd’, ‘Dy’), (‘Pr’, ‘Gd’, ‘Ho’), (‘Pr’, ‘Gd’,‘Er’), (‘Pr’, ‘Gd’, ‘Tm’), (‘Pr’, ‘Gd’, ‘Yb’), (‘Pr’, ‘Gd’, ‘Lu’),(‘Pr’, ‘Tb’, ‘Dy’), (‘Pr’, ‘Tb’, ‘Ho’), (‘Pr’, ‘Tb’, ‘Er’), (‘Pr’, ‘Tb’,‘Tm’), (‘Pr’, ‘Tb’, ‘Yb’), (‘Pr’, ‘Tb’, ‘Lu’), (‘Pr’, ‘Dy’, ‘Ho’),(‘Pr’, ‘Dy’, ‘Er’), (‘Pr’, ‘Dy’, ‘Tm’), (‘Pr’, ‘Dy’, ‘Yb’), (‘Pr’, ‘Dy’,‘Lu’), (‘Pr’, ‘Ho’, ‘Er’), (‘Pr’, ‘Ho’, ‘Tm’), (‘Pr’, ‘Ho’, ‘Yb’),(‘Pr’, ‘Ho’, ‘Lu’), (‘Pr’, ‘Er’, ‘Tm’), (‘Pr’, ‘Er’, ‘Yb’), (‘Pr’, ‘Er’,‘Lu’), (‘Pr’, ‘Tm’, ‘Yb’), (‘Pr’, ‘Tm’, ‘Lu’), (‘Pr’, ‘Yb’, ‘Lu’),(‘Nd’, ‘Sm’, ‘Eu’), (‘Nd’, ‘Sm’, ‘Gd’), (‘Nd’, ‘Sm’, ‘Tb’), (‘Nd’, ‘Sm’,‘Dy’), (‘Nd’, ‘Sm’, ‘Ho’), (‘Nd’, ‘Sm’, ‘Er’), (‘Nd’, ‘Sm’, ‘Tm’),(‘Nd’, ‘Sm’, ‘Yb’), (‘Nd’, ‘Sm’, ‘Lu’), (‘Nd’, ‘Eu’, ‘Gd’), (‘Nd’, ‘Eu’,‘Tb’), (‘Nd’, ‘Eu’, ‘Dy’), (‘Nd’, ‘Eu’, ‘Ho’), (‘Nd’, ‘Eu’, ‘Er’),(‘Nd’, ‘Eu’, ‘Tm’), (‘Nd’, ‘Eu’, ‘Yb’), (‘Nd’, ‘Eu’, ‘Lu’), (‘Nd’, ‘Gd’,‘Tb’), (‘Nd’, ‘Gd’, ‘Dy’), (‘Nd’, ‘Gd’, ‘Ho’), (‘Nd’, ‘Gd’, ‘Er’),(‘Nd’, ‘Gd’, ‘Tm’), (‘Nd’, ‘Gd’, ‘Yb’), (‘Nd’, ‘Gd’, ‘Lu’), (‘Nd’, ‘Tb’,‘Dy’), (‘Nd’, ‘Tb’, ‘Ho’), (‘Nd’, ‘Tb’, ‘Er’), (‘Nd’, ‘Tb’, ‘Tm’),(‘Nd’, ‘Tb’, ‘Yb’), (‘Nd’, ‘Tb’, ‘Lu’), (‘Nd’, ‘Dy’, ‘Ho’), (‘Nd’, ‘Dy’,‘Er’), (‘Nd’, ‘Dy’, ‘Tm’), (‘Nd’, ‘Dy’, ‘Yb’), (‘Nd’, ‘Dy’, ‘Lu’),(‘Nd’, ‘Ho’, ‘Er’), (‘Nd’, ‘Ho’, ‘Tm’), (‘Nd’, ‘Ho’, ‘Yb’), (‘Nd’, ‘Ho’,‘Lu’), (‘Nd’, ‘Er’, ‘Tm’), (‘Nd’, ‘Er’, ‘Yb’), (‘Nd’, ‘Er’, ‘Lu’),(‘Nd’, ‘Tm’, ‘Yb’), (‘Nd’, ‘Tm’, ‘Lu’), (‘Nd’, ‘Yb’, ‘Lu’), (‘Sm’, ‘Eu’,‘Gd’), (‘Sm’, ‘Eu’, ‘Tb’), (‘Sm’, ‘Eu’, ‘Dy’), (‘Sm’, ‘Eu’, ‘Ho’),(‘Sm’, ‘Eu’, ‘Er’), (‘Sm’, ‘Eu’, ‘Tm’), (‘Sm’, ‘Eu’, ‘Yb’), (‘Sm’, ‘Eu’,‘Lu’), (‘Sm’, ‘Gd’, ‘Tb’), (‘Sm’, ‘Gd’, ‘Dy’), (‘Sm’, ‘Gd’, ‘Ho’),(‘Sm’, ‘Gd’, ‘Er’), (‘Sm’, ‘Gd’, ‘Tm’), (‘Sm’, ‘Gd’, ‘Yb’), (‘Sm’, ‘Gd’,‘Lu’), (‘Sm’, ‘Tb’, ‘Dy’), (‘Sm’, ‘Tb’, ‘Ho’), (‘Sm’, ‘Tb’, ‘Er’),(‘Sm’, ‘Tb’, ‘Tm’), (‘Sm’, ‘Tb’, ‘Yb’), (‘Sm’, ‘Tb’, ‘Lu’), (‘Sm’, ‘Dy’,‘Ho’), (‘Sm’, ‘Dy’, ‘Er’), (‘Sm’, ‘Dy’, ‘Tm’), (‘Sm’, ‘Dy’, ‘Yb’),(‘Sm’, ‘Dy’, ‘Lu’), (‘Sm’, ‘Ho’, ‘Er’), (‘Sm’, ‘Ho’, ‘Tm’), (‘Sm’, ‘Ho’,‘Yb’), (‘Sm’, ‘Ho’, ‘Lu’), (‘Sm’, ‘Er’, ‘Tm’), (‘Sm’, ‘Er’, ‘Yb’),(‘Sm’, ‘Er’, ‘Lu’), (‘Sm’, ‘Tm’, ‘Yb’), (‘Sm’, ‘Tm’, ‘Lu’), (‘Sm’, ‘Yb’,‘Lu’), (‘Eu’, ‘Gd’, ‘Tb’), (‘Eu’, ‘Gd’, ‘Dy’), (‘Eu’, ‘Gd’, ‘Ho’),(‘Eu’, ‘Gd’, ‘Er’), (‘Eu’, ‘Gd’, ‘Tm’), (‘Eu’, ‘Gd’, ‘Yb’), (‘Eu’, ‘Gd’,‘Lu’), (‘Eu’, ‘Tb’, ‘Dy’), (‘Eu’, ‘Tb’, ‘Ho’), (‘Eu’, ‘Tb’, ‘Er’),(‘Eu’, ‘Tb’, ‘Tm’), (‘Eu’, ‘Tb’, ‘Yb’), (‘Eu’, ‘Tb’, ‘Lu’), (‘Eu’, ‘Dy’,‘Ho’), (‘Eu’, ‘Dy’, ‘Er’), (‘Eu’, ‘Dy’, ‘Tm’), (‘Eu’, ‘Dy’, ‘Yb’),(‘Eu’, ‘Dy’, ‘Lu’), (‘Eu’, ‘Ho’, ‘Er’), (‘Eu’, ‘Ho’, ‘Tm’), (‘Eu’, ‘Ho’,‘Yb’), (‘Eu’, ‘Ho’, ‘Lu’), (‘Eu’, ‘Er’, ‘Tm’), (‘Eu’, ‘Er’, ‘Yb’),(‘Eu’, ‘Er’, ‘Lu’), (‘Eu’, ‘Tm’, ‘Yb’), (‘Eu’, ‘Tm’, ‘Lu’), (‘Eu’, ‘Yb’,‘Lu’), (‘Gd’, ‘Tb’, ‘Dy’), (‘Gd’, ‘Tb’, ‘Ho’), (‘Gd’, ‘Tb’, ‘Er’),(‘Gd’, ‘Tb’, ‘Tm’), (‘Gd’, ‘Tb’, ‘Yb’), (‘Gd’, ‘Tb’, ‘Lu’), (‘Gd’, ‘Dy’,‘Ho’), (‘Gd’, ‘Dy’, ‘Er’), (‘Gd’, ‘Dy’, ‘Tm’), (‘Gd’, ‘Dy’, ‘Yb’),(‘Gd’, ‘Dy’, ‘Lu’), (‘Gd’, ‘Ho’, ‘Er’), (‘Gd’, ‘Ho’, ‘Tm’), (‘Gd’, ‘Ho’,‘Yb’), (‘Gd’, ‘Ho’, ‘Lu’), (‘Gd’, ‘Er’, ‘Tm’), (‘Gd’, ‘Er’, ‘Yb’),(‘Gd’, ‘Er’, ‘Lu’), (‘Gd’, ‘Tm’, ‘Yb’), (‘Gd’, ‘Tm’, ‘Lu’), (‘Gd’, ‘Yb’,‘Lu’), (‘Tb’, ‘Dy’, ‘Ho’), (‘Tb’, ‘Dy’, ‘Er’), (‘Tb’, ‘Dy’, ‘Tm’),(‘Tb’, ‘Dy’, ‘Yb’), (‘Tb’, ‘Dy’, ‘Lu’), (‘Tb’, ‘Ho’, ‘Er’), (‘Tb’, ‘Ho’,‘Tm’), (‘Tb’, ‘Ho’, ‘Yb’), (‘Tb’, ‘Ho’, ‘Lu’), (‘Tb’, ‘Er’, ‘Tm’),(‘Tb’, ‘Er’, ‘Yb’), (‘Tb’, ‘Er’, ‘Lu’), (‘Tb’, ‘Tm’, ‘Yb’), (‘Tb’, ‘Tm’,‘Lu’), (‘Tb’, ‘Yb’, ‘Lu’), (‘Dy’, ‘Ho’, ‘Er’), (‘Dy’, ‘Ho’, ‘Tm’),(‘Dy’, ‘Ho’, ‘Yb’), (‘Dy’, ‘Ho’, ‘Lu’), (‘Dy’, ‘Er’, ‘Tm’), (‘Dy’, ‘Er’,‘Yb’), (‘Dy’, ‘Er’, ‘Lu’), (‘Dy’, ‘Tm’, ‘Yb’), (‘Dy’, ‘Tm’, ‘Lu’),(‘Dy’, ‘Yb’, ‘Lu’), (‘Ho’, ‘Er’, ‘Tm’), (‘Ho’, ‘Er’, ‘Yb’), (‘Ho’, ‘Er’,‘Lu’), (‘Ho’, ‘Tm’, ‘Yb’), (‘Ho’, ‘Tm’, ‘Lu’), (‘Ho’, ‘Yb’, ‘Lu’),(‘Er’, ‘Tm’, ‘Yb’), (‘Er’, ‘Tm’, ‘Lu’), (‘Er’, ‘Yb’, ‘Lu’), and (‘Tm’,‘Yb’, ‘Lu’).
 11. The method of claim 1, wherein the step of measuringthe immobile trace elements in a soil sample after application of themineral amendment further comprises lysing the soil sample andapplication of ethylene diaminetetraacetic acid.
 12. A mineral amendmentfor enhanced rock weathering comprising: an agglomerated silicatemineral having an average particle size of about 0.5 mm to about 3 mm,the agglomerated silicate mineral comprising a comminuted silicatemineral and a binder; and immobile trace elements, the immobile traceelements comprising one or more rare earth elements, rare metals, andtransition metals.
 13. The mineral amendment of claim 12, wherein: therare earth elements comprise scandium (Sc), yttrium (Y), lanthanum (La),cerium (Ce), praseodymium (Pr), neodymium (Nd), samarium (Sm), europium(Eu), gadolinium (Gd), terbium (Tb), dysprosium (Dy), holmium (Ho),erbium (Er), thulium (Tm), ytterbium (Yb), and lutetium (Lu); the raremetals comprise beryllium (Be), cesium (Cs), gallium (Ga), germanium(Ge), hafnium (Hf), niobium (Nb), rubidium (Rb), tantalum (Ta), titanium(Ti), vanadium (V), and zirconium (Zr); and the transition metalscomprise nickel (Ni), chromium (Cr), and zinc (Zn).
 14. The mineralamendment of claim 12, wherein the immobile trace elements are enrichedin abundance compared to their abundance in naturally found silicatemineral.
 15. The mineral amendment of claim 12, wherein the comminutedsilicate mineral is forsterite (Mg₂SiO₄) or blast furnace slag.
 16. Themineral amendment of claim 12, wherein the comminuted silicate mineralhas an average particle size between about 80 μm and about 100 μm. 17.The mineral amendment of claim 12 further comprises a slow-releaseacidifier.
 18. The mineral amendment of claim 17, wherein theslow-release acidifier comprises one or more of gibbsite (Al(OH)₃) orgypsum (CaSO₄)-2H₂O).
 19. The mineral amendment of claim 12 furthercomprises one or more of zinc, a nutrient, and a biological-derivedcomponent.
 20. The mineral amendment of claim 12 is applied to soil toenhance rock weathering.
 21. The soil of claim 20, wherein the amount ofCO₂ removed by the mineral amendment can be verified by measuring theimmobile trace elements.